The sharp exponent for the minimal distance problem

Technical Summary: The Sharp Exponent for the Minimal Distance Problem

Problem Statement
The paper addresses the minimal distance problem for point-line incidence configurations. Given $n$ points $x_1, \dots, x_n$ in the unit square $[0, 1]^2$ and $n$ lines $\ell_1, \dots, \ell_n$ such that each point $x_i$ lies on its corresponding line $\ell_i$, the problem seeks to maximize the minimum distance between any point $x_i$ and any line $\ell_j$ where $i \neq j$. This quantity is denoted as:
$$\Delta_{PL}(n) := \sup \min_{i \neq j} \text{dist}(x_i, \ell_j)$$
Prior to this work, the asymptotic behavior of $\Delta_{PL}(n)$ was bounded by $n^{-1/2} \lesssim \Delta_{PL}(n) \lesssim n^{-2/3+o(1)}$. The upper bound was established by Cohen, the author, and Zakharov [7], while the best known lower bound was $\Delta_{PL}(n) \gtrsim n^{-\gamma}$ with $\gamma \approx 0.8107$, derived from constructions related to the Furstenberg–Sárkőzy problem [11]. The gap between the exponent $2/3$ (upper bound) and $\approx 0.81$ (lower bound) remained open.

Methodology
The paper constructs a family of point-line configurations that achieve the lower bound matching the known upper bound. The methodology evolves from a previous construction by Hunter, the author, Verstraëte, and Zhang [11] but introduces a critical shift from integer-based sets to algebraic number fields.

1. The Induced-Matching Identity: The core algebraic mechanism relies on the polynomial $Q(x, y) = 2x - y^2$. For a point $p=(x,y)$, the line $L_p$ tangent to the parabola $Q=Q(p)$ at $p$ is defined. A key identity shows that if a point $p'$ lies on $L_p$, then $Q(p) - Q(p')$ must be a perfect square. Consequently, if the set of $Q$-values is square-difference-free (no two distinct elements differ by a perfect square), the only incidence between the set of points and their associated tangent lines is the trivial one ($p$ on $L_p$).

2. From Integers to Number Fields: Previous constructions used subsets of integers $A \subset [N]$ with no square differences (related to the Furstenberg–Sárkőzy problem). The density of such sets in integers is limited by current bounds on the Furstenberg–Sárkőzy problem, preventing the exponent from reaching $2/3$.
   To bypass this, the author constructs the set $A$ within the ring of integers $\mathcal{O}_K$ of a totally real number field $K$ of high degree $d$.

   • Trace-Zero Lattice: Instead of a Ruzsa-like set, the construction utilizes the trace-zero slice of the ring of integers: $\Lambda_0^K = \{a \in 2\mathcal{O}_K : \text{Tr}_{K/\mathbb{Q}}(a) = 0\}$.
   • Square-Difference-Free Property: In a totally real field, the trace of a non-zero square is strictly positive ($\text{Tr}_{K/\mathbb{Q}}(z^2) > 0$ for $z \neq 0$). Therefore, the difference of any two distinct elements in the trace-zero hyperplane cannot be a square. This guarantees the square-difference-free property required for the induced matching.

3. Geometric Embedding: The algebraic points are mapped to the Euclidean plane $[0, 1]^2$ via a specific real embedding $\sigma: K \to \mathbb{R}$ and an anisotropic scaling map $\Phi_{M,N}$. The separation distance in the Euclidean plane is controlled by the field norm of the algebraic difference $D(p, p')$. By choosing a field of sufficiently high degree $d$, the "dilution" of the fixed codimension (trace-zero) allows the separation exponent to approach the theoretical limit.

Key Contributions and Results

• Theorem 1.1: For every $\epsilon > 0$, there exist arbitrarily large $n$ and configurations of $n$ point-line pairs such that $\text{dist}(x_i, \ell_j) \ge n^{-2/3-\epsilon}$ for all $i \neq j$.
• Corollary 1.2: The sharp exponent for the minimal distance problem is exactly $2/3$. Specifically, $\Delta_{PL}(n) = n^{-2/3+o(1)}$ as $n \to \infty$.
• Resolution of the Gap: The result closes the gap between the upper bound $n^{-2/3+o(1)}$ (from [7]) and the previous lower bound.

Significance and Claims
The paper claims to have solved the minimal distance problem by determining the correct asymptotic exponent.

• Independence from Furstenberg–Sárkőzy: The proof bypasses the limitations of the Furstenberg–Sárkőzy problem (specifically the Ruzsa limit of $\beta \approx 0.733$) by utilizing the algebraic structure of high-degree totally real number fields. This demonstrates that the density constraints of the integer case do not apply in the same way to the geometric realization in higher-degree fields.
• Heilbronn Triangle Problem: The author notes that while $\Delta_{PL}(n)$ is connected to the Heilbronn triangle problem (via the relation $\Delta_H(n) \lesssim n^{-1/2}\Delta_{PL}(n)$), the sharpness of the $2/3$ exponent for $\Delta_{PL}(n)$ does not necessarily imply the sharpness of the current best upper bound for the Heilbronn problem ($n^{-7/6+o(1)}$). The author conjectures that the Heilbronn bound can be further improved, as the reduction from Heilbronn to the minimal distance problem discards significant geometric information.
• Furstenberg–Sárkőzy Implications: The work also suggests that the minimal distance problem is linked to the Furstenberg–Sárkőzy problem via a reverse inequality. However, the paper does not claim to improve the bounds for the Furstenberg–Sárkőzy problem itself; rather, it uses the number field construction to optimize the geometric problem.

Acknowledgments
The author explicitly acknowledges that the decisive idea of using the codimension-one trace-zero lattice to upgrade the exponent from $4/5$ to the sharp $2/3$ was generated by OpenAI's GPT-5.6 Sol Pro. The author's original intent was to use high-degree number fields to reach the Ruzsa endpoint ($4/5$), but the specific construction using the trace-zero slice to achieve the sharp exponent was an AI-driven insight.